IV - Convergence and Summability Theorems

Abstract

This chapter discusses elementary convergence criteria for Fourier series. The convergence property of a Fourier series at a point x depends only on the function in a neighborhood of x. This is the so-called local property of convergence. The Fourier coefficients are determined by the values of the function over the whole interval (—π, π), but the convergence at x depends only on the values of the function around a point x. The chapter also discusses convergence of Fourier–Stieltjes series, Fourier's integral theorems, inversion formulas for Fourier transforms, inversion formula for Fourier–Stieltjes transforms, summability theorems for Fourier transforms, and Fourier series and approximate Fourier series of a Fourier–Stieltjes transform.